El Problema de Lambert en el Contexto de Optimización de Trayectorias Espaciales

Esta tesis se basa en el estudio de la trayectoria que pasa por dos puntos en el problema de los dos cuerpos, inicialmente desarrollado por Lambert, del que toma su nombre. En el pasado, el Problema de Lambert se ha utilizado para la determinación de órbitas a partir de observaciones astronómicas de los cuerpos celestes. Actualmente, se utiliza continuamente en determinación de órbitas, misiones planetaria e interplanetarias, encuentro espacial e interceptación, o incluso en corrección de orbitas. Dada su gran importancia, se decide investigar especialmente sobre su solución y las aplicaciones en las misiones espaciales actuales. El campo de investigación abierto, es muy amplio, así que, es necesario determinar unos objetivos específicos realistas, en el contexto de ejecución de una Tesis, pero que sirvan para mostrar con suficiente claridad el potencial de los resultados aportados en este trabajo, e incluso poder extenderlos a otros campos de aplicación. Como resultado de este análisis, el objetivo principal de la Tesis se enfoca en el desarrollo de algoritmos para resolver el Problema de Lambert, que puedan ser aplicados de forma muy eficiente en las misiones reales donde aparece. En todos los desarrollos, se ha considerado especialmente la eficiencia del cálculo computacional necesario en comparación con los métodos existentes en la actualidad, destacando la forma de evitar la pérdida de precisión inherente a este tipo de algoritmos y la posibilidad de aplicar cualquier método iterativo que implique el uso de derivadas de cualquier orden. En busca de estos objetivos, se desarrollan varias soluciones para resolver el Problema de Lambert, todas ellas basadas en la resolución de ecuaciones transcendentes, con las cuales, se alcanzan las siguientes aportaciones principales de este trabajo: • Una forma genérica completamente diferente de obtener las diversas ecuaciones para resolver el Problema de Lambert, mediante desarrollo analítico, desde cero, a partir de las ecuaciones elementales conocidas de las cónicas (geométricas y temporal), proporcionando en todas ellas fórmulas para el cálculo de derivadas de cualquier orden. • Proporcionar una visión unificada de las ecuaciones más relevantes existentes, mostrando la equivalencia con variantes de las ecuaciones aquí desarrolladas. • Deducción de una nueva variante de ecuación, el mayor logro de esta Tesis, que destaca en eficiencia sobre todas las demás (tanto en coste como en precisión). • Estudio de la sensibilidad de la solución ante variación de los datos iniciales, y como aplicar los resultados a casos reales de optimización de trayectorias. • También, a partir de los resultados, es posible deducir muchas propiedades utilizadas en la literatura para simplificar el problema, en particular la propiedad de invariancia, que conduce al Problema Transformado Simplificado. ABSTRACT This thesis is based on the study of the two-body, two-point boundary-value problem, initially developed by Lambert, from who it takes its name. Since the past, Lambert's Problem has been used for orbit determination from astronomical observations of celestial bodies. Currently, it is continuously used in orbit determinations, for planetary and interplanetary missions, space rendezvous, and interception, or even in orbit corrections. Given its great importance, it is decided to investigate their solution and applications in the current space missions. The open research field is very wide, it is necessary to determine specific and realistic objectives in the execution context of a Thesis, but that these serve to show clearly enough the potential of the results provided in this work, and even to extended them to other areas of application. As a result of this analysis, the main aim of the thesis focuses on the development of algorithms to solve the Lambert’s Problem which can be applied very efficiently in real missions where it appears. In all these developments, it has been specially considered the efficiency of the required computational calculation compared to currently existing methods, highlighting how to avoid the loss of precision inherent in such algorithms and the possibility to apply any iterative method involving the use of derivatives of any order. Looking to meet these objectives, a number of solutions to solve the Lambert’s Problem are developed, all based on the resolution of transcendental equations, with which the following main contributions of this work are reached: • A completely different generic way to get the various equations to solve the Lambert’s Problem by analytical development, from scratch, from the known elementary conic equations (geometrics and temporal), by providing, in all cases, the calculation of derivatives of any order. • Provide a unified view of most existing relevant equations, showing the equivalence with variants of the equations developed here. • Deduction of a new variant of equation, the goal of this Thesis, which emphasizes efficiency (both computational cost and accuracy) over all other. • Estudio de la sensibilidad de la solución ante la variación de las condiciones iniciales, mostrando cómo aprovechar los resultados a casos reales de optimización de trayectorias. • Study of the sensitivity of the solution to the variation of the initial data, and how to use the results to real cases of trajectories’ optimization. • Additionally, from results, it is possible to deduce many properties used in literature to simplify the problem, in particular the invariance property, which leads to a simplified transformed problem.

Testing for Subsidiarity. Bruges European Economic Policy (BEEP) Briefing 13/2006

In the EU circuit (especially the European Parliament, the Council and Coreper) as well as in national parliaments of the EU Member States, one observes a powerful tendency to regard 'subsidiarity' as a 'political' issue. Moreover, subsidiarity is frequently seen as a one-way street : powers going 'back to' Member States. Both interpretations are at least partly flawed and less than helpful when looking for practical ways to deal with subsidiarity at both EU and Member states' levels. The present paper shows that subsidiarity as a principle is profoundly 'functional' in nature and, hence, is and must be a two-way principle. A functional subsidiarity test is developed and its application is illustrated for a range of policy issues in the internal market in its widest sense, for equity and for macro-economic stabilisation questions in European integration. Misapplications of 'subsidiarity' are also demonstrated. For a good understanding, subsidiarity being a functional, two-way principle neither means that elected politicians should not have the final (political!) say (for which they are accountable), nor that subsidiarity tests, even if properly conducted, cannot and will not be politicised once the results enter the policy debate. Such politicisation forms a natural run-up to the decision-making by those elected for it. But the quality and reasoning of the test as well as structuring the information in a logical sequence ( in accordance with the current protocol and with the one in the constitutional treaty) is likely to be directly helpful for decisionmakers, confronted with complicated and often specialised proposals. EU debates and decision-making is therefore best served by separating the functional subsidiarity test (prepared by independent professionals) from the final political decision itself. If the test were accepted Union-wide, it would also assist national parliaments in conducting comparable tests in a relatively short period, as the basis for possible joint action (as suggested by the constitutional treaty). The core of the paper explains how the test is formulated and applied. A functional approach to subsidiarity in the framework of European representative democracy seeks to find the optimal assignment of regulatory or policy competences to the various tiers of government. In the final analysis, this is about structures facilitating the highest possible welfare in the Union, in the fundamental sense that preferences and needs are best satisfied. What is required for such an analysis is no less than a systematic cost/benefit framework to assess the (de)merits of (de)centralisation in the EU.

Selected major elements and trace elements of ferrar dolerite clasts from sediment core CRP-1 (Table 1,2)

One of the objectives of the Cape Roberts Project is to study the tectonic history of the western Ross Sea region. Timing of the uplift of the Transantarctic Mountains, which are adjacent to the drillsite, will be a component of the tectonic studies (International Steering Committee, 1994; Cale Roberts Science Team, 1998a). The study of the clast samples from the core will be an important means of providing insight into the timing of uplift of the Transantarctic Mountains. Tholeiitic igneous rocks of the Jurassic (180 Ma) Ferrar large igneous province (FLIP) are widespreaded along the Transantarctic Mountains and have the potential to provide distinct indicators of erosion during uplift of the mountains. In the Transantarctic Mountains adjacent to the Cape Roberts drill site the FLIP is represented by lavas and pyroclastic of the Kirkpatrick basalts and by thick Ferrar dolerite sills which intrude the Beacon Supergroup sediments and, occasionally, the granitic basement rocks. In the Prince Albert Mountains, the youngest Kirkpatrick basalt lava is over 150 m thick, and has a very distinct high TiO2 chemical composition which is unique in the FLIP. If such rocks can be identified in the core they may provide precise timing of the initiation of uplift and denudation of the Transantarctic Mountains. Here we report on an examination of 20 Ferrar dolerite clasts. This brief report is intended as a pilot study to the examination of FLIP clasts from older drillcore.

Report of Machine Workshop on the 6-GeV Synchrotron Radiation Source : National Bureau Standards, Gaithersburg, Maryland, March 28-30, 1985.

"Sponsored by the US Department of Energy through the Advanced Photon Source Steering Committee"--Pref.

Wisconsin Library network plan report : recommendations for network development at the area level /

WI docs. no: Ed.3/2:2250

"Streamlined auditing and monitoring of community based services: first steps toward a more efficient system for providers, state government, and the community" : report to the Illinois General Assembly as required by PA 96-1141 /

The Steering Committee recognizes that the Departments may need to implement reviews necessary to address specific Federal requirements and that while consolidated reporting should be used when possible, the review and reporting processes should facilitate the ability to garner Federal funding. However, the analysis thus far confirms the need for change and a significant potential to reduce redundant monitoring and reporting. At the end of this Executive Summary, there is a summary presentation, including a timeline and progress indicators, that gives more details on these recommendations.

Some proposals to management : a suggested approach to sound industrial and public relations /

Mode of access: Internet.

Some proposals to management. A suggested approach to sound industrial and public relations.

Mode of access: Internet.

Status report[s] on Clinch River Study.

UC-70

Towards complete and accurate reporting of studies of diagnostic accuracy: the STARD initiative

Objective To improve the accuracy and completeness of reporting of studies of diagnostic accuracy, to allow readers to assess the potential for bias in a study, and to evaluate a study's generalisability. Methods The Standards for Reporting of Diagnostic Accuracy (STARD) steering committee searched the literature to identify publications on the appropriate conduct and reporting of diagnostic studies and extracted potential items into an extensive list. Researchers, editors, and members of professional organisations shortened this list during a two day consensus meeting, with the goal of developing a checklist and a generic flow diagram for studies of diagnostic accuracy. Results The search for published guidelines about diagnostic research yielded 33 previously published checklists, from which we extracted a list of 75 potential items. At the consensus meeting, participants shortened the list to a 25 item checklist, by using evidence, whenever available. A prototype of a flow diagram provides information about the method of patient recruitment, the order of test execution, and the numbers of patients undergoing the test under evaluation and the reference standard, or both. Conclusions Evaluation of research depends on complete and accurate reporting. If medical journals adopt the STARD checklist and flow diagram, the quality of reporting of studies of diagnostic accuracy should improve to the advantage of clinicians, researchers, reviewers, journals, and the public.

Application of Regularized Discriminant Analysis

2000 Mathematics Subject Classification: 62H30, 62P99